Universal weight systems from a minimal ℤ₂²-graded Lie algebra

2024/10/08 by N. Aizawa, Aizawa, N., Daichi Kimura +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT)

paper · pdf · doi:10.48550/arxiv.2410.05845

openalex publication_date 2024/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Color Lie algebras, which were introduced by Ree, are a graded extension of Lie (super)algebras by an abelian group. We show that the color Lie algebras can be used to construct universal weight systems for knot invariants of of Vassiliev and Kontsevich. As a simple example, we take ℤ₂ × ℤ₂ as the grading group and consider the four-dimensional color Lie algebra called A1_ε. The weight system constructed from A1_ε is studied in some detail and some relations between the weights, such as the recurrence relation for chord diagrams, are derived. These relations show that the weight system from A1_ε is a hybrid of those from sl(2) and gl(1|1).

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